Problem: What is the inverse of the function $h(x)=\dfrac{-2x-1}{x+5}$ ? $ h^{-1}(x) =$
Answer: Let's start by replacing $h(x)$ with $y$. $y=\dfrac{-2x-1}{x+5}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{-2y-1}{y+5}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{-2y-1}{y+5}&=x \\\\ -2y-1&=x(y+5) \\\\ -2y-1&=xy+5x \\\\ -2y-xy&=5x+1 \\\\ y(-2-x)&=5x+1 \\\\ y&=\dfrac{5x+1}{-2-x} \end{aligned}$ In conclusion, this is the inverse function: $h^{-1}(x)=\dfrac{5x+1}{-2-x}$ [I saw someone solve this problem by originally solving for x. Were they wrong?]